Quantized Decay Charges: Mechanisms and Models

• Quantized decay charges are discrete observables arising when decay processes select or suppress charge quantization in mesoscopic, pump, and non-Hermitian systems.
• In mesoscopic conductors, experiments reveal a universal square-root collapse of charge visibility near ballistic channels and exponential thermal suppression.
• In pump and non-Hermitian systems, decay cascades and master equation approaches quantify captured charge and spatial decay profiles, linking integer invariants to system connectivity.

“Quantized decay charges” does not designate a single invariant across the literature. In mesoscopic transport, it can denote the controlled decay of Coulomb-blockade charge quantization under quantum and thermal fluctuations, with a visibility $\Delta Q$ that plays the role of an order parameter (Jezouin et al., 2016). In tunable-barrier pumps, it refers to the discrete trapped charge selected by a backtunneling decay cascade during the decoupling stage (Kashcheyevs et al., 2014). In non-Hermitian problems, the phrase has been used both for an integer controlling the mean decay time, $\langle T\rangle=w/(2\Gamma)$, and for node-wise integer or half-integer decay charges in directed graphs that support pure exponential decay modes (Thiel et al., 2019, Liu et al., 15 Jul 2025). Related work further connects charge quantization to detector resolution, compact electric-flux sectors, higher-order boundary charge, and quantized plasmonic surface-charge decay.

1. Principal meanings and taxonomy

Across these works, the shared structure is that a charge-like observable is either selected by a decay process or loses its discreteness in a controlled way. The underlying objects, however, are different: island charge in a Coulomb-blockaded conductor, the captured occupation of a pump quantum dot, a non-Hermitian decay invariant, a graph-theoretic node charge, or a compact electric flux.

Framework	Quantized object	Representative relation
Metallic island with QPCs	Visibility/order parameter $\Delta Q$	$\Delta Q \propto \sqrt{1-\tau}$, with thermal suppression $\propto \exp(-\pi^2 k_B T/E_C)$ (Jezouin et al., 2016)
Tunable-barrier electron pump	Captured charge $\langle n\rangle$	Decay-cascade fit with $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$ (Kashcheyevs et al., 2014)
Single-channel non-Hermitian decay	Mean decay time	$\langle T\rangle = w/(2\Gamma)$ (Thiel et al., 2019)
Directed non-Hermitian graph	Node decay charge $Q_\alpha$	$$Q_\alpha = (\#\text{OE}_\alpha-\#\text{IE}_\alpha)/2$$ (Liu et al., 15 Jul 2025)
Massive Schwinger model on a circle	Electric field sector	$\langle T\rangle=w/(2\Gamma)$0 (Hu et al., 2021)

A persistent source of ambiguity is that some papers study the quantization of a quantity that governs decay, whereas others study the decay of charge quantization itself. The mesoscopic-island problem is of the latter type; the non-Hermitian and pump problems are of the former.

2. Decay of charge quantization in mesoscopic conductors

In “Controlling charge quantization with quantum fluctuations” a hybrid single-electron transistor is formed from a micrometer-scale metallic island connected through two tunable quantum point contacts, with charging energy $\langle T\rangle=w/(2\Gamma)$1 and electrostatic spectrum $\langle T\rangle=w/(2\Gamma)$2 (Jezouin et al., 2016). The experimentally monitored quantity is the visibility

$\langle T\rangle=w/(2\Gamma)$3

where $\langle T\rangle=w/(2\Gamma)$4 corresponds to deep Coulomb blockade and $\langle T\rangle=w/(2\Gamma)$5 to effectively continuous charge. Near the ballistic limit, the same paper identifies $\langle T\rangle=w/(2\Gamma)$6 with the amplitude of the island charge oscillations and treats it as an order parameter for how much charge quantization survives.

The central low-temperature result is a universal square-root collapse of this order parameter as a ballistic channel is approached. In the asymmetric quantum regime $\langle T\rangle=w/(2\Gamma)$7, $\langle T\rangle=w/(2\Gamma)$8,

$\langle T\rangle=w/(2\Gamma)$9

with $\Delta Q$0 and $\Delta Q$1. More generally, the data show that for $\Delta Q$2, $\Delta Q$3 for many $\Delta Q$4, and that a single ballistic channel is sufficient to destroy charge quantization. The paper identifies $\Delta Q$5 as a ballistic critical point: $\Delta Q$6 there and remains indistinguishable from zero once one channel is ballistic.

Thermal fluctuations add an independent suppression. For an isolated island, the oscillatory part of the partition function is already suppressed as $\Delta Q$7, and the same exponential factor survives in the strongly connected device. In the strong-thermal regime the visibility obeys

$\Delta Q$8

while the experiment found an empirical form with coefficient $\Delta Q$9 in the exponent over the explored temperature range. The resulting picture is precise: quantum fluctuations produce a non-analytic square-root decay in residual reflection, while thermal fluctuations produce an exponential decay in $\Delta Q \propto \sqrt{1-\tau}$0 (Jezouin et al., 2016).

3. Decay cascades and captured charge in tunable-barrier pumps

In “Modeling of a tunable-barrier non-adiabatic electron pump beyond the decay cascade model,” the relevant charge is the number of electrons left on a quantum dot after the entrance barrier is raised and before ejection to the drain (Kashcheyevs et al., 2014). The pumped current is

$\Delta Q \propto \sqrt{1-\tau}$1

so the decoupling stage determines the quantized transferred charge. During that stage, backtunneling to the source competes with the time-dependent rise of the dot levels. Quantized capture is therefore a decay-selection problem: the cascade $\Delta Q \propto \sqrt{1-\tau}$2 preferentially removes excess electrons until a low-$\Delta Q \propto \sqrt{1-\tau}$3 state freezes out.

The original decay-cascade description yields the double-exponential fit

$\Delta Q \propto \sqrt{1-\tau}$4

with $\Delta Q \propto \sqrt{1-\tau}$5 in the ideal zero-temperature limit. The generalized treatment replaces this with sequential-tunneling master equations for $\Delta Q \propto \sqrt{1-\tau}$6, explicit time-dependent rates $\Delta Q \propto \sqrt{1-\tau}$7, time-dependent electrochemical potentials $\Delta Q \propto \sqrt{1-\tau}$8, and finite-temperature detailed balance. A key parameterization is

$\Delta Q \propto \sqrt{1-\tau}$9

where $\propto \exp(-\pi^2 k_B T/E_C)$0 is the plunger-to-barrier ratio.

The main refinement is that the plateau-shape parameter becomes

$\propto \exp(-\pi^2 k_B T/E_C)$1

so it contains both a temporal separation of decay rates and an energetic separation through the addition energy $\propto \exp(-\pi^2 k_B T/E_C)$2. Finite temperature introduces repopulation from the source lead. The crossover is controlled by $\propto \exp(-\pi^2 k_B T/E_C)$3: for $\propto \exp(-\pi^2 k_B T/E_C)$4, the decay-cascade formula remains accurate, while for $\propto \exp(-\pi^2 k_B T/E_C)$5 an alternative empirical ansatz,

$\propto \exp(-\pi^2 k_B T/E_C)$6

fits better, with $\propto \exp(-\pi^2 k_B T/E_C)$7 (Kashcheyevs et al., 2014). In this setting, “quantized decay charges” refers to integer-like captured charge emerging from state-dependent decay and re-population dynamics.

4. Non-Hermitian decay invariants in time and on directed graphs

A different meaning appears in “Quantization of the mean decay time for non-Hermitian quantum systems.” There the system evolves with

$\propto \exp(-\pi^2 k_B T/E_C)$8

and, for the initial state $\propto \exp(-\pi^2 k_B T/E_C)$9, the mean decay time is quantized: $\langle n\rangle$0 The integer $\langle n\rangle$1 is the number of distinct energy levels of $\langle n\rangle$2 whose eigenspaces have nonzero overlap with the decay state, and it is also the winding number of a transform of the resolvent. Close to parameter values where $\langle n\rangle$3 changes, the decay-time distribution develops rare long-lived events; finite observation windows can then report an apparent smaller $\langle n\rangle$4 than the asymptotic value (Thiel et al., 2019).

In “Quantized decay charges in non-Hermitian networks characterized by directed graphs,” the quantized object is instead local in real space (Liu et al., 15 Jul 2025). The systems are directed graphs with non-reciprocal hopping that support pure decay modes, i.e. eigenstates with strictly monotone exponential profiles,

$\langle n\rangle$5

without the oscillatory factors characteristic of conventional non-Hermitian skin modes. For a node $\langle n\rangle$6, the decay charge is defined by

$\langle n\rangle$7

with the sum over connected nodes. For the graph class constructed in the paper, this simplifies to

$\langle n\rangle$8

The charges are therefore integers or half-integers, satisfy $\langle n\rangle$9, and depend only on directed connectivity. The paper derives universal conditions for such pure decay modes, including a binary Hamiltonian constraint $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$0, and reports microwave resonant-circuit experiments that reproduce the predicted straight-line $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$1 profiles (Liu et al., 15 Jul 2025).

These two non-Hermitian constructions quantify different observables: a global temporal invariant $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$2 and local spatial decay charges $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$3. A plausible implication is that “quantized decay charge” has become a useful umbrella only after specification of which decay observable is meant.

5. Detector resolution and interaction-induced loss of emitted charge quantization

“Charge quantization and detector resolution” argues that whether charge appears quantized or continuous is not solely an intrinsic property of the system but depends on the spatial resolution of the detector (Riwar, 2020). For a sharply defined region,

$$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$4

has integer eigenvalues, whereas a coarse-grained detector measures

$$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$5

which need not. In Josephson-junction arrays in the superinductor regime, the detector is modeled by weights $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$6 over junction currents, and the geometric current-correlation sum obeys

$$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$7

A sharp detector gives $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$8, while a maximally fuzzy detector drives the sum toward $$\delta_2=\ln(\Gamma_2/\Gamma_1)+E_c/\Delta_{\text{ptb}}$$9. The same circuit can therefore support a $\langle T\rangle = w/(2\Gamma)$0-periodic, quantized-Cooper-pair description for a sharp detector and an effectively continuous quasicharge description for a fuzzy one (Riwar, 2020).

A complementary mechanism appears in “Interaction effects and charge quantization in single-particle quantum dot emitters,” where the issue is not detector coarseness but dot-edge Coulomb coupling (Wagner et al., 2018). After an exact mapping to the spin-boson model, the emitted current is

$\langle T\rangle = w/(2\Gamma)$1

with

$\langle T\rangle = w/(2\Gamma)$2

Even if the dot fully discharges, the downstream wave packet carries $\langle T\rangle = w/(2\Gamma)$3, not the bare charge $\langle T\rangle = w/(2\Gamma)$4. For repulsive interactions, $\langle T\rangle = w/(2\Gamma)$5. The paper therefore concludes that Coulomb interactions “lead to a destruction of precise charge quantization in the emitted wave-packet” and cast doubts on the viability of this setup as a source of quantized charge pulses (Wagner et al., 2018).

Taken together, these works rule out a naive identification between microscopic carrier discreteness and measured or emitted quantized charge. Detector basis, spatial coarse-graining, and unavoidable interactions can each suppress or reinterpret what counts as a quantized decay charge.

6. Broader generalizations: compact flux, boundary charges, Heun quantization, and quantized plasmon decay

Several additional works use closely related ideas in mathematically distinct settings. In “Electric Field Decay Without Pair Production,” the massive Schwinger model on a compact spatial circle shows that when the circumference satisfies $\langle T\rangle = w/(2\Gamma)$6, generic initial electric fields are stable, whereas fields quantized as

$\langle T\rangle = w/(2\Gamma)$7

oscillate in time from $\langle T\rangle = w/(2\Gamma)$8 to $\langle T\rangle = w/(2\Gamma)$9, with exponentially small probability of taking other values (Hu et al., 2021). The instanton analysis yields novel straight-line, lemon, and related worldline saddles, so the decay of flux becomes a tunneling problem between discrete sectors rather than ordinary Schwinger pair production.

In “Fractional hinge and corner charges in various crystal shapes with cubic symmetry,” the quantized charges are static boundary invariants rather than dynamical decay observables (Naito et al., 2021). For cubic-symmetry polyhedra, hinge and corner charges are fixed only modulo shape-dependent fractions because boundary relaxation can shift their integer parts. A particularly sharp example is the appearance of a fractionally quantized corner charge $Q_\alpha$0 mod $Q_\alpha$1 in a truncated cube or truncated octahedron, and the paper shows that irreducible representations at high-symmetry momenta are insufficient to determine the corner charge without an additional Wilson-loop invariant $Q_\alpha$2.

“Quantization of the charge in Coulomb plus harmonic potential” uses “charge” in yet another sense: the Coulomb coupling $Q_\alpha$3 in a radial Schrödinger problem with potential $Q_\alpha$4 becomes quantized together with the energy because polynomial biconfluent-Heun solutions require two truncation conditions (Choun et al., 2020). The allowed pairs $Q_\alpha$5 are discrete, the charge parameter is state dependent, and two radial quantum numbers are required. This suggests a different route to “quantized decay charges,” namely through state-dependent couplings that would change discretely under transitions between bound states.

Two further generalizations broaden the conceptual landscape. “Semi-classical description of electrostatics and quantization of electric charge” proposes that electric charge must be a rational multiple of $Q_\alpha$6, and that boundary conditions forcing a field wavefunction to vanish on specific surfaces quantize the corresponding source charge density (Bhattacharya, 2023). “Field Quantization for Radiative Decay of Plasmons in Finite and Infinite Geometries” quantizes surface charge-density oscillations themselves: each plasmon mode is a bosonic harmonic oscillator with Hamiltonian of the form $Q_\alpha$7, and radiative decay rates are derived by coupling these quantized surface-charge modes to the photon field (Bagherian, 2019). In the latter setting, the “charges” are collective surface-charge excitations whose decay is literally the annihilation of a quantized charge-density mode.

These extensions show that the topic spans mesoscopic transport, non-Hermitian topology, compact gauge dynamics, higher-order crystalline charge, semiclassical electrostatics, and plasmonics. What remains common is the coexistence of two ingredients: a discrete charge-like structure and a decay, tunneling, emission, or coarse-graining mechanism that reveals, suppresses, or redefines that discreteness.